Fully Kinetic Modeling of a Divergent Cusped-Field Thruster
by
Stephen R. Gildea
Submitted to the Department of Aeronautics and Astronautics
in partial fulfillment of the requirements for the degree of
Master of Science in Aeronautics and Astronautics
MASSACHUSETTS INSTFIUTE
OF TECHNOLOGY
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
OCT 13 2009
LIBRARIES
September 2009
@ Massachusetts Institute of Technology 2009. All rights reserved.
ARCHIVES
Author ........
Department of Aeronautics and Astronautics
August 20, 2009
A
z
Certified by...
Manuel Martinez-Sanchez
Professor
Thesis Supervisor
Accepted by ..............
David L. Darmofal
Professor
Associate Department Head
Chair, Committee on Graduate Students
Fully Kinetic Modeling of a Divergent Cusped-Field Thruster
by
Stephen R. Gildea
Submitted to the Department of Aeronautics and Astronautics
on August 20, 2009, in partial fulfillment of the
requirements for the degree of
Master of Science in Aeronautics and Astronautics
Abstract
A fully kinetic, particle-in-cell plasma simulation tool has been incrementally developed
by members of the Massachusetts Institute of Technology Space Propulsion Laboratory.
Adapting this model to simulate the performance and plasma dynamics of a divergent
cusped-field thruster is discussed. Strong magnetic fields in the cusps (B0.5 T) necessitate using a time step on the order of a picosecond in order to resolve electron cyclotron
trajectories. As a result, successfully completing a divergent cusped-field thruster simulation with the full magnetic field strength has yet to be accomplished. As an intermediate
step, simulation results of a divergent cusped-field thruster with the magnetic field at 1/5
the actual value are presented, including performance parameters and internal plasma structure details. Evidence suggests that even at 1/5 the magnetic field strength, ions are fully
magnetized within certain regions of the divergent cusped-field thruster. This has strong
implications concerning the basic operating principles of the thruster because the Hall effect
does not result in a net flow of current in regions where ions are fully magnetized. Further
modifications that may lead to successful simulations of divergent cusped-field thrusters at
full magnetic field strength are also outlined, which may allow for more detailed studies of
the plasma structure and performance of the cusped-field thruster.
Thesis Supervisor: Manuel Martinez-Sanchez
Title: Professor
Acknowledgments
I would like to thank my advisor, Professor Martinez-Sanchez, for always making himself
available for discussion. Also, this project could not have been completed without the
valuable insights of Dr. Oleg Batishchev. I want to thank him for continually providing me
with new perspectives on what it means to execute and interpret a kinetic simulation. Dr.
James Szabo and Dr. Justin Fox were also helpful when the only people left to ask were
the authors of the code.
I would also like to thank the administrators and everyone involved with the Science
Mathematics and Research for Transformation Scholarship Program. This program has
truly been a catalyst for exciting scientific opportunities.
Above all, I would like to thank my wife, Erin, for her unending support, strength and
love.
Contents
1
2
Introduction to Divergent Cusped-Field Thrusters
........................
1.1
Cusped-Field Thrusters
1.2
Divergent Cusped-Field Thrusters
..................
Introduction to Fully Kinetic Modeling
.....................
2.1
Modeling Plasma Thrusters .
2.2
Feasible Implementation of the Kinetic Method ...........
2.2.1
Reducing the Problem Size and Accelerating Convergence .
23
3 DCFT Modeling Results
3.1
.
23
. . .
.
24
. . . .
.
24
. . . .
.
24
...................
. . . .
.
25
....................
. . . .
.
27
. . . . . . . . . .
27
..................
3.1.1
Grid Generation
3.1.2
Calculating the Magnetic Field . . . . . . . ...
. . .
.........................
3.2
Collisions
3.3
Coding Modifications . .
3.4
Simulation Results . .
3.4.1
4
. . . .
Boundary Conditions and Simulation Inputs . . . . . . .
Results with One-Fifth Magnetic Field Strength
45
Conclusions and Future Work
4.1
4.2
DCFT Operating Principles .......
.........
. . . .
.
45
4.1.1
Electron and Ion Drift Velocities . . . . . . . ..
. . . .
.
46
4.1.2
Ion and Electron Magnetization . . . . . . . ...
. . . .
.
47
..... .
51
Future Work
...
.......................
List of Figures
1-1
Two DCFT visualizations.
One shows the DCFT firing, the other is an
engineering model of an assembled DCFT . ..................
14
3-1
Boundary conditions used in the present investigation. . ............
24
3-2
The 2 dimensional, structured, non-orthogonal, variable mesh used in these
simulations ..................
3-3
..........
.......
25
Magnetic field vector and magnitude for the simulation results shown in this
paper.
The actual magnetic field is a factor of 5 larger than shown.
A
simulated magnetic field was calculated using the Ansoft Maxwell software
package[1, 2], and then interpolated to fit the computational mesh shown in
Figure 3-2.
......
........
.................
26
3-4
The number of super-particles for neutrals, electrons and ions.
3-5
Spatial variation of the number of super-particles of charged species at the
3 5 0 , 0 0 0 th
iteration. ...................
. .....
............
.
29
..
34
3-6
Simulated performance varialbes: Isp, thrust, anode current and beam current. 35
3-7
Electron and Ion Temperature
...................
3-8
Ion velocity components for the
3 5 0 , 0 0 0 th
3-9
Electron velocity components for the
......
36
iteration. . ..............
3 5 0 , 0 0 0 th
37
iteration. . ........
. .
38
3-10 Electron temperature is plotted in components parallel and perpendicular to
the magnetic field.
...................
............
..
39
3-11 Ion temperature is plotted in components parallel and perpendicular to the
magnetic field.
...................
..............
3-12 Electron and ion number density distributions for the
3 5 0 , 0 0 0 th
..
iteration.
40
41
3-13 The potential distribution with plasma present in direct comparison with the
vacuum potential distribution.
...................
......
42
43
3-14 Axial and radial electric field components. . ..................
4-1
Calculated ExB drift speeds and a value proportional to gradient-curvature
drifts.
4-2
.....
..............
...............
.
48
Azimuthal current density, calculated from quantities shown previously in
Figures 3-9(c) and 3-12(a).
. .....
......
..........
.
49
4-3 DCFT Ion Magnetization Assessment: Ion Hall Parameter, calculated as the
ratio of ion cyclotron frequency to ion collision frequency. The collisional
cross sections used to calculate the ion collision frequency were: Ion-Neutral
Charge Exchange, Ion-Neutral Elastic Scattering, and Ion-Electron Coulomb.
The effects of wall recombination collisions were ignored. Including ion wall
collisions would reduce the Hall parameter close to the wall. These results
are from a simulation completed with the magnetic field strength reduced by
a factor of 5. ......
.............
...............
4-4 Average electron and ion Larmor radii ....................
50
. . .
52
List of Tables
3.1
A listing of the collisions represented in the model. Some collision types that
are not included are also provided. ...................
....
25
12
Chapter 1
Introduction to Divergent
Cusped-Field Thrusters
1.1
Cusped-Field Thrusters
Stationary plasma type (SPT) Hall thrusters have been operated in earth orbit since 1972[3]
with typical efficiencies near 50% and a specific impulse of 1600 s[4]. However, the operational lifetime of SPT Hall thrusters is often limited by erosion of dielectric material used to
provide insulation from the plasma[5]. Researchers at the Massachusetts Institute of Technology (MIT) Space Propulsion Laboratory (SPL) began investigating divergent cuspedfield (DCF) thrusters[1, 2, 6] to determine their promise as a longer lifetime alternative to
existing low power Hall thrusters. Other cusped-field thrusters include the High Efficiency
Multi-stage Plasma (HEMP) thrusters patented by THALES Electron Devices[7, 8, 9] and
the Cylindrical Hall Thruster (CHT) developed at the Princeton University Electric Propulsion & Plasma Dynamics Laboratory[10]. A detailed comparison of the DCFT design to
the HEMP, CHT, and other cusped-field devices is provided by Courtney et al.[6].
1.2
Divergent Cusped-Field Thrusters
DCF thrusters are electric propulsion devices characterized by a divergent, dielectric wall
channel and a magnetic topology dominated by two magnetic cusps and a magnetic bottle
near the anode (see Figure 3-3). The magnetic circuit consists of three annular samarium
cobalt (SmCo) permanent magnets arranged such that like poles face one another, and a
I
(a) DCFT firing.[1]
~
------
~
(b) 3D Engineering Model of DCFT with an
angled section.[2]
Figure 1-1: DCFT Visualizations.
core at the base of the thruster composed of 1018 carbon steel (see Figure 1-1(b)). Xenon
gas is injected 5 mm upstream from the downstream face of the anode through a porous,
stainless steel disc embedded in the magnetic core. In this way, the neutral injection and
anode locations are decoupled from one another in the Divergent Cusped-Field Thruster
(DCFT) design. For complete details of the DCFT and existing data, please consult previous studies[2, 1, 6, 11]. Existing experimental data[2, 1, 6] catalogues DCFT performance
(anode specific impulse, thrust, anode thrust efficiency) at operating conditions ranging
between 100 W to 500 W, with a nominal operating condition corresponding to a xenon
flow rate of 8.5 sccm and an applied anode voltage of 550 V. At this operating point, the
DCFT consumes 242 W of power, and produces 13.4 mN of thrust while operating with an
anode thrust efficiency of 44.5%. Although basic DCFT operating principles are thought to
include azimuthal electron currents within regions characterized by radial magnetic fields,
the competing roles that strong magnetic bottles and internal electric fields play in determining the location and efficiency of ionization (whether in the upstream cusp, downstream
cusp, or both) have yet to be explicitly measured in experiments or rigourously explained
based on theoretical arguments or simulation results. Demonstrating the effectiveness of the
magnetic topology and divergent chamber geometry as erosion barriers remains unachieved
as well. The goal of the present investigation is to gain a deeper understanding of underlying mechanisms at play within cusped-field thrusters using a fully kinetic, particle-in-cell
computational model of the DCFT. This simulation tool will be benchmarked against detailed DCFT experimental measurements (completed by members of the MIT SPL[11], Air
Force Research Laboratory and Stanford Plasma Physics Laboratory). A working model
will help evaluate how design decisions will affect performance metrics such as plume divergence angle, ionization efficiency, ion energy efficiency, and the spatially resolved energy
flux to DCFT chamber surfaces. Figure 1-1(a) shows the first successful firing of the DCFT
at the MIT SPL.
16
Chapter 2
Introduction to Fully Kinetic
Modeling
Fully kinetic modeling refers to representing a fluid as discrete particles whose trajectories
are dynamically simulated. Specifying the type (mass, charge, etc.) and state (position and
velocity) of each particle at a given instant in time allows the state of each particle to be
determined at any future time by integrating the equation of motion for each particle under
the influence of known forces. Knowledge of the state of each member in a set of particles
allows the velocity distribution function for each species to be tabulated at each location
in the simulation domain. The velocity distribution function links the view of a fluid as
a discrete set of particles to that of a continuum; providing the density, bulk velocity and
temperature of each species as moments of the velocity distribution function.
The present work considers electromagnetic forces acting on neutral particles, ions and
electrons. Additionally, the trajectories of particles are strongly influenced by collisions
with other particles and boundaries. The details of how these forces and interactions are
accounted for are provided in Chapter 3, and references cited throughout.
2.1
Modeling Plasma Thrusters
Modeling plasma systems is an ongoing effort of research throughout the scientific community. However, the following narrative focuses on the development of models for Hall
Thrusters within the Space Propulsion Laboratory at the Massachusetts Institute of Technology, as they are by far the most relevant.
In 1995, Fife provided a simulation tool to model the performance and plasma dynamics
of Hall Thrusters[12]. Neutrals and ions were modeled as particles within a two-dimensional
(R-Z plane), nonorthogonal, structured mesh (hence, Particle-in-Cell).
Conversely, the
electron velocity distribution function was everywhere assumed to be Maxwellian and the
electrons were governed by fluid equations. The model was refined and compared to experimental measurements in 1999[13].
Due to errors attributed to the assumption of an isotropic electron velocity distribution function, Szabo expanded the Particle-in-Cell (PIC) approach to include electrons in
2001[14]. Although the spatial dimensionality of the model remained the same, velocities of
all particles were accounted for in three dimensions (2D3V). Later, the model was modified
to include dielectric boundaries[15] and used to conduct an investigation into wall erosion
on high powered Hall Thrusters[16].
In order to make implementations more tractable for higher powered thrusters, Fox
parallelized the method in 2007[17], allowing for the growing availability of parallel computing hardware to be exploited.
Fox added the additional capability of the analytical
quench model[18, 19, 20] in an attempt to self-consistently account for anomalous electron
diffusion[21]. Additional features that Fox added to the code, such as an adaptive meshing
capability, are not present in the version used in this investigation.
2.2
Feasible Implementation of the Kinetic Method
The computational resources necessary to perform a complete, kinetic simulation are well
beyond the capabilities of contemporary technology. The DCFT will be used as an example
to justify this assertion:
* On the order of 1013 particles need to be simulated for an assumed average number
density of 1012 per ccm. For a 2D3V simulation, that means integrating 5 x 1013 scalar
differential equations each time step!
* Maintaining numerical stability requires the largest allowable time step to be the
inverse of the largest characteristic frequency in the domain. Using the previously
assumed number density, and a maximum magnetic field of .53 T[2], the time steps
based on the electron plasma and cyclotron frequencies are on the order of 1 ps and
10 ps, respectively. Simulating the residence time of a neutral Xenon population with
s
a temperature of 1000 K (DCFT chamber length is about 5 cm) requires about 2 x 10
iterations.
* For a plasma number density on the order of 1012 per ccm and an electron temperature
of 20 eV, a representative electron Debye length for the DCFT is 30 pim. Resolving the
Debye length, a necessary condition for numerical stability, requires a mesh with about
1010 nodes. Therefore, determining the potential distribution for a single iteration
requires the solution of a system with approximately 1010 unknowns.
Even under the assumption that each action above (integrating one equation of motion
or solving the potential of a single node) could be accomplished with a single floating point
operation, and that using additional processors produces an idealized speed up factor of
(1/n), the time required to run the simulation on a parallel machine with 100, 1 GHz
processors would still be over 3000 years 1 ! Fortunately, Szabo developed a methodology[14]
that vastly reduces the necessary computational resources while still providing credible
simulation results.
2.2.1
Reducing the Problem Size and Accelerating Convergence
Szabo's model modifies selected fundamental physical quantities associated with plasma
systems in order to reduce the computational scale of the problem, and then takes additional
steps to convert quantities in the artificial system to those that would be measured in a
real plasma. Full details of Szabo's method are presented in Reference [14], with the most
daring assumption being that the bulk quantities of temperature (T) and number density
2
(n) remain unaffected by the changes outlined in the remainder of this chapter .
Increasing Free Space Permittivity: -y
Equation 2.1 shows how the permittivity of free space (E) is modified within the simulation.
The value of -y is specified in an input file by the user.
_ y2Eo
0E*
(2.1)
1
This estimate also assumes the machine has enough RAM to store the entire simulation as it runs.
2
Consequently, T* -- T and n* - n . The superscript * will be used throughout to indicate simulated
values of a quantity, whereas variables lacking the asterisk would be the 'real' value.
Increasing the free space permittivity has two beneficial consequences related to reducing
the enormity of the required simulation task.
Equations 2.2 and 2.3 illustrate how the
length and time scales that govern the behavior of the simulation are both increased by the
factor y.
= ~(AD,e)
Ake =
e2 ne
S
p,e
(2.2)
(2.3)
An important caveat is that the simulation time scale will only increase if the modified
electron plasma frequency continues to be the largest characteristic frequency in the domain.
For instance, the electron cyclotron frequency is unchanged by any of the altered quantities
in Szabo's method, meaning a reduction in the electron plasma frequency by a certain factor
will eventually cause the electron cyclotron frequency to dominate. At this point, increasing
'y will not lengthen the allowable timestep any further. Strong magnetic fields present in
the DCFT cause the electron cyclotron frequency to dominate for most values of gamma
used in this investigation.
Additionally, increasing the simulated Debye length allows the number of nodes required
to represent a fixed domain size to decrease by a factor of
7y2 . For statistical reasons, an
increase in the simulated Debye length also allows the simulation to be run using y2 fewer
particles for identical levels of statistical accuracy per cell.
Reducing the Mass of Heavy Species
Even with the effects of the permittivity factor y, the time scales of heavy (neutrals and
ions) and light (electrons) particles remain very different. In order to encourage the heavy
particles to converge more quickly, the masses of all neutral and ionic species are reduced
by the factor f, as shown in Equation 2.4. Simulated speeds of particles are now a factor
of Vf larger than the actual value, for equivalent energies.
mH-
S
rH
f
,
f > 1
(2.4)
Super-Particles
Finally, in order for the model to run within the constraint of available random access
memory, super-particles are simulated, each one representing - 108 real particles in DCFT
results presented here. Currently, ion and electron super-particles are equally weighted,
though neutral super-particles are even larger. Ideally, this is done to bring the number of
simulated particles for each species to the same order of magnitude. However, the drawback associated with increasing the size of super-particles is the loss of resolution that occurs
when representing moments of the velocity distribution function. For instance, the smallest
increment of density that can be simulated is a single particle per cell. Alternatively, simulated fluxes, such as the anode current or thrust, are subject to the same phenomenon within
the simulation. The step size of a simulated flux value corresponds to one super-particle
per time step. With such a strong magnetic field present in the DCFT, the combination of
large super-particles and a time step dominated by the electron cyclotron frequency only
adds to the statistical noise associated with quantities such as thrust, anode current and
beam current.
Altering the mass and size of heavy species also changes simulated density and mean
free path values. However, this is overcome by reducing the mass flow rate by a factor
v,
while increasing collisional cross sections by the same factor. Additional details of this
methodology are presented in Szabo's Ph.D. thesis[14], such as reducing the magnetic field
seen by ions by the factor
fV
to prevent ions from becoming falsely magnetized.
22
Chapter 3
DCFT Modeling Results
3.1
Boundary Conditions and Simulation Inputs
The first steps toward adapting the PIC model to the DCFT were to provide the vector
magnetic field (Figure 3-3), structured mesh (Figure 3-2), and boundary types (Figure 31) as input files.
The magnetic field was calculated using the Ansoft Maxwell software
package, and then interpolated to the structured mesh. No significant differences between
the calculated and interpolated magnetic fields have been observed, and the agreement
between the simulated and measured magnetic field magnitudes along the centerline of
the magnetic configuration has been demonstrated previously[2]. Figure 3-1 illustrates the
different boundary types as classified by the PIC model. The area labeled as SIMULATION
REGION represents the allowed locations of all particles in the simulation.
The right
hand side of the diagram shows the exit plane of the thruster. Future simulations would
benefit from expanding this boundary further into the plume. The ANODE POTENTIAL
represents all regions that have a constant potential equal to the applied voltage from
anode to cathode, with the cathode assumed to be at zero potential. Explicitly modeling
an external, off-axis cathode is not possible under the assumption of azimuthal symmetry.
As a cathode surrogate, the upper-rightmost node of the simulation region is specified to be
at cathode potential for all iterations. Additionally, electrons are injected at a temperature
between 0.2 and 2.5 eV along cells on the righthand side boundary in order to prevent any
deficit of electrons from developing.
CENTER LINE
DIELECTRIC (BN)
SIMULUATION REGION l
ANODE POTENTIAL M
Figure 3-1: Boundary conditions used in the present investigation.
3.1.1
Grid Generation
using an elliptic mesh generator
The grid representing the simulation domain was made
shown in Figure 3-2 correspond to
created by Fife[12]. The node locations of the mesh
mesh should represent the physical
engineering models of the DCFT. The computational
should be less than the modified electron
domain as closely as possible, and the node spacing
Debye length (see Section 2.2.1) on the entire domain.
3.1.2
Calculating the Magnetic Field
Maxwell software package, and then
The magnetic field was calculated using the Ansoft
in Figure 3-3. No significant
interpolated to the structured mesh. The result is shown
magnetic fields have been noticed.
differences between the calculated and interpolated
3.2
Collisions
the model[14].
Table 3.1 lists the types of collisions that are included in
2.5
E
M:
2
1.5
1
0,5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Z (cm)
Figure 3-2: The 2 dimensional, structured, non-orthogonal, variable mesh used in these
simulations.
Table 3.1: A listing of the collisions represented in the model. Some collision types that are
not included are also provided.
Collision Types Included
(e-)-(n) Elastic Scattering
(e-)-(n) Ionization
(e-)-(n) Excitation
(e-)-(e-) Coulomb
(e )-(i+,++) Coulomb
(i+'++)-(n) Charge Exchange
(i+ '+ + )-(n) Scattering
(i+ '+ + )-(e - ) Wall Recombination
Collision Types Not Included
(i+,++)-(i + ,+ + ) Coulomb
(n)-(n) Scattering
(e-)-(i+ ' + + ) Bulk Recombination
3.3
Coding Modifications
Changing the model to accurately simulate particle interactions with the new DCFT boundary conditions has been the most challenging aspect of this project. The most significant
2.5
1.5
0.5
Z (cm)
Figure 3-3: Magnetic field vector and magnitude for the simulation results shown in this
paper. The actual magnetic field is a factor of 5 larger than shown. A simulated magnetic
field was calculated using the Ansoft Maxwell software package[1, 2], and then interpolated
to fit the computational mesh shown in Figure 3-2.
modifications include updating the logical structure of the code to include sloped walls,
and tailoring the logical sheath method first implemented by Blateau[15] to the DCFT. To
speed up execution time for cases when the electron cyclotron frequency determines the
time step, a scheme is now executed that delays the poisson solver to run on the time scale
of the maximum inverse electron plasma frequency, rather than during each iteration. The
poisson solver represents a third to a half of the total computational cost per iteration,
meaning that significant speedup of the model is obtained, and some of the original benefits
associated with increasing the free space permittivity are recuperated. Finally, the method
by which a plasma is originally seeded into the domain was varied to determine if the observed plasma structure was in fact a characteristic of the DCFT, or just an artifact of the
initialization approach.
initialization approach.
3.4
Simulation Results
The procedure for running a PIC simulation is as follows. First, a simulation is run with
only neutral particles to allow an equilibrated neutral background on which to seed a plasma
to develop. Next, electrons and singly charged ions are seeded throughout the domain at
a specified temperature and density. From this point on, the particle trajectories are determined by the local Lorentz force, algorithms governing interactions with other particles
and physical boundaries (ionization, wall recombination, collisions, reflection), and by the
quench rule - with its roll in enhancing cross field diffusion above classical levels where appropriate. The collection of simulated particles evolves forward in time for a preset number
of iterations, after which the results are analyzed. Although a fully kinetic simulation can
never reach a truly steady state, the PIC model is said to have converged when characteristic simulated values obtain relatively constant values, or when outputs become periodic in
nature. Quantities of interest for judging whether a simulation has converged or not include
the total number of super-particles for each species present in the simulation domain, anode
specific impulse, thrust, anode current and other flux related quantities.
3.4.1
Results with One-Fifth Magnetic Field Strength
Convergence
An important step toward creating a useful model of DCF thrusters is creating a model
that converges. Reducing the magnetic field by a factor of 5 decreases the gyrofrequencies
of all simulated particles by an identical factor, reducing the number of time steps required
1
to simulate a given interval of time (i.e. the approximate neutral residence time) . In this
manner, a simulation of the DCFT was completed, providing convergent results for various
performance variables and plasma characteristics. Unfortunately, the data presented here
can not be directly compared to existing experimental results, though existing data may
suggest the plausibility of the following results. The B/5 simulation was run with a xenon
volumetric flow rate of 8.5 sccm and an applied potential difference from anode to cathode of
450 V. The mass factor value f=1000, and the permittivity factor -=1200 were used. While
the mass factor is of the same order as values used in previous works, the permittivity factor
1Assuming the electron gyrofrequency remains larger than the modified electron plasma frequency. Modifying the electron plasma frequency is done by artificially increasing the permittivity of free space. See
Section 2.2.1.
is much larger than has been used for past studies at the MIT SPL. The super-particle size
for electrons and ions was 1.3x108, while neutral particles began the simulation 50 times
larger than charged particles.
Figures 3-5(a), 3-5(b), 3-6(a), 3-6(b) and 3-6(c) all suggest convergence of the DCFT
model. Unfortunately, no such convergence has been obtained for a simulation incorporating
the full magnetic field strength present in a real DCF thruster. The simplest explanation
is that the simulation has not been allowed to run for a sufficient number of iterations,
though full strength B cases have shown divergent characteristics less than half way through
attempts to run the model for a million iterations or more. Another possibility is that during
the strong transient behavior exhibited by the converged simulation during the first 250,000
to 300,000 iterations (as seen in Figures 3-5(a) and 3-5(b)) the simulation diverges due to
high plasma densities (corresponding to very large numbers of super-particles). This would
occur if local Debye lengths were to fall below the characteristic length of a given cell for
a prolonged number of iterations. It is possible that this instability is initiated within
one portion of the grid, and then spreads to surrounding cells. In fact, this behavior was
observed to arise in the upper right corner of the anode, even in the B/5 case, which led
to simulations being run with y=1 2 0 0 , rather than with -y < 100 as before. Conceivably,
the full B case exacerbates the problem in which the local Debye length falls well below the
resident cell size, leading to instabilities which cause the simulation to diverge after several
tens of thousands of iterations. All the results shown in the next section correspond to a
simulation run with the magnetic field reduced to 1/5 its actual value.
Figures 3-5(a) and 3-5(b) show how the total number of super-particles for each species
changes over the course of the entire simulation. After 300,000 iterations, the number of
super-particles for each species levels off, indicating that the simulation has converged. Also
significant is the fact that after about the
2 5 0 , 0 0 0th
iteration, the total numbers of electron
and ion super-particles are within 10% of each other, indicating that the simulation domain
is nearly globally neutral. The slight excess of electron super-particles can be explained
by the fact that, while ions can only enter the simulation domain as a result of ionization
events, electrons have the additional source of the righthand-side boundary, where electrons
are injected to implicitly simulate the presence of a cathode. Another explanation could be
that the ion super-particle population includes a number of doubly charged ions, requiring
a fewer number of simulated ions compared to simulated electrons.
.......
.......
.......
.....
.......
......
.......
......
4.oE.05
4.5E-.6
4.OE.06
' .E*05
_____
los
.
E 31.5E406
S20E.05
ilaos
0
21.OE06
II
z
IIIIIIIIIII~~~~I~~~
0
I
~
I
~ ~ ~ ~
300000
400000
,
5.0E-04
5.0E.05
100000
200000
300000
~I
400000
500000
000000
Iteration Number
(a) The number of neutral species super-particles
for each iteration.
200000
50000
Iteration Number
(b) The number of charged species super-particles
for each iteration. The count of ion super-particles
includes doubly and singly charged ions.
Figure 3-4: The number of super-particles representing each species of particle is plotted
for each iteration of the simulation.
Super-particle Distribution
Figures 3-5(a) and 3-5(b) show the distribution of super-particles for electrons and ions as
the number of super-particles per cell. As expected, electron super-particle densities are the
greatest within the magnetic cusps. As a result, the values of simulated electron macroscopic
quantities such as density, temperature and velocity are relatively insensitive to the flux of
a small number of particles into or out of cells within the cusps. Contrastingly, the values of
such quantities in regions outside the cusps can be expected to exhibit stronger statistical
fluctuations about a mean value. Less obvious is the presence of a large group of simulated
electrons just above the anode. Apparently, a strong electric field, arising from the sharp
corner of the downstream anode face, results in the creation of a very dense plasma. This
feature is present in many results presented here.
Simulated ions are are much more spread out than the electrons. As will be addressed
when Figures 3-12(a) and 3-12(b) are presented, the computational grid used here limits the
value of the permittivity factor (-y) to a minimum value of around 1200. Thus, the strong
internal electric fields that would normally maintain local charge neutrality are lessened as
y increases.
Performance Variables
Figure 3-6(a) illustrates that the anode specific impulse of the modeled DCFT converges to a
value of approximately 1500 s, or within 8% of the value measured experimentally (1400 s)
for the same operating flow rate and anode potential[1].
Although making comparisons
between experimental data and simulated data for the same thruster with different magnetic
field strengths is not ideal, it is the only rubric by which the simulated results can be
measured. As such, they should not be taken absolutely. However, this result suggests that
a converged simulation for the full strength magnetic field case should be able to predict
specific impulse to within a reasonable margin. According to Figure 3-6(b), the average
total simulated thrust is w14 mN, the beam current is
.6 A, the anode current is -1 A,
and the neutral current (corresponding to the current that would result from singly ionizing
all incoming propellent) is 0.61 A. Thus, according to Equations 3.1, 3.2 and 3.3, rt = 26%,
?]b = 60% and u, = 98%.
T
2
rt -2aPa
=ib
77 =
Ib=
a
(3.1)
(3.2)
(3.3)
In
The anode thrust, beam and utilization efficiencies could be interpreted to mean, that at
1/5 B, the DCFT is operating in the high current mode characterized in existing DCFT
references[1, 6, 2]. This is supported by Figure 3-7(a), which shows two distinct regions of
relatively high temperature electrons between the anode and the upstream cusp, and again
between the two cusps. The high utilization efficiency is probably due to the fact that the
weak magnetic field allows the ionization region to spread closer to the walls than otherwise,
allowing a greater portion of the flow to be ionized. In addition, doubly charged ions could be
contributing to the high utilization efficiency determined by the model. Another symptom
of the weaker magnetic field is the low anode thrust efficiency and beam efficiency. The 1/5
magnetic field reduction increases electron mobility to the anode by weakening the magnetic
bottle system that would normally be in place. The result is a larger anode current, reducing
the beam efficiency and increasing the power consumed by pumping electrons from anode
to cathode.
Plasma Structure
The ion velocity vectors as visualized in Figures 3-8(a), 3-8(b) and 3-8(c) suggest that
the magnetic field has a strong influence on ion trajectories.
Figure 3-8(c) shows bulk
azimuthal ion speeds increasing near the walls, which are the areas closest to the strongest
magnetic fields. The directions of bulk ion speeds in the azimuthal direction are consistent
with grad-B and curvature drifts. Furthermore, the upstream cusp has a more distinct
effect on ion velocities than the downstream cusp, especially as seen in Figure 3-8(a). For
both cusps, the ions are seen to obtain a maximum axial speed just before entering the
cusp, and then emerge from the cusped-field region with a speed lower than they entered
with. Unpublished measurements of ion velocities using laser-induced fluorescence on a
CHT reveal azimuthal ion speeds with magnitudes of the same order as shown in Figure
3-8(c)(Hargus, W.A., personal communication, July 23, 2009). An examination of ions and
their velocities outside the chamber is not possible due to the location of the downstream
mesh boundary. When interpreting these or any plots in this section, it is important to
keep in mind that these are snapshots of a kinetic simulation, and so these values do not
necessarily represent time averaged quantities very accurately. From Figures ?? and ??,
these fluctuations will be lower within regions with larger numbers of simulated particles
per cell. Also, the imposed boundary conditions on the righthand-side boundary may be
altering the ion trajectories in a manner that is not physical. The artificial nature of the
exit plane boundary conditions should be considered when interpreting these data.
Figures 3-9(a), 3-9(b) and 3-9(c) show the axial, radial and azimuthal macroscopic
electron velocity magnitudes, respectively. The axial speed distribution shows the electron
velocities slowing within each cusp on their way upstream toward the anode. Figure 39(b) predicts that the only locations where the bulk flow is traveling toward the divergent,
insulating wall are within the cusps and just above the anode. Azimuthally, the direction
of electron velocities do not appear to be consistent with ExB drifts seen in Hall thrusters.
In Section 4.1.1, an attempt to account for the direction and magnitude of polar electron
velocity is made.
Figures 3-10(a), 3-10(b), 3-11(a) and 3-11(b) were calculated from simulation results at
the
3 5 0 ,0 0 0 th
iteration according to Equations 3.4 and 3.5. These equations hold if only
hydrostatic pressure terms are considered to contribute to the mean squared random ve-
locity magnitude. The parallel and perpendicular directions are defined relative the local
direction of the magnetic field. The electron temperature plots demonstrate that the electron temperatures in the parallel and perpendicular directions are very similar in value,
suggesting that random electron energy distributions are nearly isotropic with respect to
magnetic field lines. The same cannot be said of ions, however, as their temperature within
the cusps is nearly entirely perpendicular to the magnetic field, while between cusps, the
ion temperature is nearly all parallel to magnetic field lines.
) + TR
Ti ~ Tz
TL
1
2
-
T ±Tz(
2
B
)+T(
B
(3.4)
)
B
B
2
(3.5)
Figures 3-12(a) and 3-12(b) show an instantaneous view of the electron and ion number
density distributions for the
3 5 0 , 0 0 0 th
iteration. This iteration number was chosen because
it is representative of values that can be expected after the simulation was observed to
converge. Figures 3-13(a) and 3-7(a) are also snapshots of the respective spatially varying
variables at the
3 5 0 ,0 0 0 th
iteration. Apparent regions lacking charge neutrality can be
explained by the fact that electrostatic forces have been reduced by a factor 72, where 7 =
1200 in this case. However, the boundary potentials of the anode and cathode are enforced,
meaning that as the permittivity factor y -
oc, the potential distribution that results from
solving Poisson's equation is replaced by a linear potential distribution associated with
Laplace's equation, as shown in Equation 3.6.
lim
lim V 2 0 = -y-'
-y-*oo
Pcharge = 0
6,Y2
(3.6)
From examining Figure 3.4.1, it seems likely that the neutral injector floating boundary
is initialized to a zero potential and then remains at that potential for the duration of
the simulation. This will be addressed in future simulations. Figure 3-12(a) very clearly
illustrates the fact that electrons are effectively bottled in and between the cusps. The
high electron density in the cusps is to be expected, due to the fact that electron guiding
centers are slowed as they approach the ceramic wall. Therefore, the electron density should
increase. Conversely, as the electrons follow the magnetic field lines out of the cusps, into
regions with weaker magnetic fields, their velocity parallel to the magnetic field increases
as the tradeoff necessary to maintain the invariant value of the average magnetic moment
for each electron occurs.
The components of the simulated electric field vectors are plotted in Figures 3-14(a) and
3-14(b). The electric field structure within the cusps appears to change sign in the radial
and axial directions.
Time Resolved Analysis
Time resolved anode current measurements show that DCFT operation is characterized
by low frequency oscillations on the order of 3 or 4 kHz[22]. As such, the period of such
an oscillation is on the order of 250 ,ps. The total time elapsed for the simulated data
presented here is only about 24 ps, which is not a sufficient amount of time over which to
expect time resolved currents to be reproduced, especially if you consider the percentage of
the simulated time the thruster required to converge to a performance regime (see Figure
3-5(b)).
;;;;;;; ;;;;;;;
.~;;;;
;;;;
.................
;.
100
2.5
S
E
S2
1.5
64
16
9
0
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Z(cm)
(a) The number of electron super-particles per cell.
2.5
CC
2
1.5
0.5
0.5
1
1.5
2
2.5
3.5
4
4.5
Z (cm)
(b) The number of ion super-particles per cell. The counts shown here include
singly and doubly charged ions.
Figure 3-5: The spatial variation of the concentration of simulated particles per cell at the
associated with moments of the velocity distribu3 5 0 , 0 0 0 th iteration. The statistical noise
tion function in a given cell is proportional to the inverse of the square root of the number
of particles occupying that cell. Thus, a cell containing only 4 particles would have about
4 times as much noise as a cell containing 64 particles (i.e. a standard deviation 4 times
larger).
..................................................
No w
30
Z25
E
U
2
20
0
500
1000
Normalized Time
2500
3000
'IAnode
IBeam
..
1000
2000
(b) Thrust at each instant of time. The large
amount of statistical noise apparent here is due to
the size of the ion super-particles. As the superparticle size increases, the resolution of any particle
flux deteriorates, for a fixed velocity magnitude.
(a) Anode specific impulse at each instant of time.
500
150
Iteration
500
2000
2500
3000
Normalized Time
(c) Anode and beam currents plotted at each instant of
time.
Figure 3-6: Performance variables plotted as a function of normalized time. For the present
simulation, one unit of time is 8.0 ns.
_;,..;;;,..;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
~-----------~;;;;;;;;;;;;;;;;;;;;;;;;;;
I
2.5
-ev!
40
220
0
1.5
0.5
0
1.5
1
0.5
2.5
2
3
3.5
4
4.5
Z (cm)
(a) Instantaneous electron temperature distribution at the
2.5 "
%,2 -
3 50 ,0 0 0 th
iteration.
T (eV)
40
30
1.5 -
10o
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Z (cm)
th
(b) Instantaneous ion temperature distribution at the 350,000 iteration.
Figure 3-7: Electron and ion temperature.
..........
2.5
1r" 2
1.5
0.5
0
0
1
0.5
1.5
2
2.5
Z(cm)
3
3.b
4
4.0
o
4
4.0
3.5
4
4.5
(a) Ion axial speed.
3.5
V, (m/s)
15000
13500
12000
10500
9000
7500
*
2.5
6000
4500
IT" 2
31500
7500
1.5
0.5
0
0
0.5
1
1.,
Z
Z.0
11
Z (cm)
(b) Ion radial speed.
3.5 I
(mis)
100
800
600
0.5
0
0.5
1
1.5
2
2.5
3
Z (cm)
(c) Ion azimuthal speed.
Figure 3-8: Ion speeds in all three dimensions are plotted for the
3 5 0 ,0 0 0 th
iteration.
~L1
I
~
.
...................
.
3 -
2
-.
_ - i~i..nift
;;;
MONWii~
1160000
0.
1
-320000
.
254
.
-1.52E+06
-1.76E.+C16
0.5
1.5
1
0.5
0
2.5
2
3.5
3
4
4.5
4
4.5
4
4.5
Z(cm)
(a) Electron axial speed.
2.5-
600000
1.5
1
2.5
2
3.5
3
0.5
Ee) Electron a ial speed.
S-400 (c
0
-600000
-800000
-06
1.5
0.
0.5
O
1
1.5
2
2.5
3
3.5
Zz(cm)
(cm)
(b) Electron radial speed.
a
2.5
1.5
-
0.5
Z(cm)
I
-------
;;;;;
;;;;;;;;;;; ;;;;.........................................
42
36
2.5
0
1
0.5
0
0,5
1
1.5
2
2.5
3
3.5
4
4.5
Z (cm)
(a) Instantaneous electron temperature, parallel to the magnetic field lines.
2.5
,2
n-
1.5
1
0.5
0
0.5
1
1.5
2
2.b
3
3.
4
4.
Z (cm)
(b) Instantaneous electron temperature, perpendicular to the magnetic field lines.
Figure 3-10: Electron temperature with respect to magnetic field directionality.
....
......
~
;;;;;;;;~;;;;;;;;;;;;;;;;;;;;;;;~;;;;;~;;;;;;;;~;;;;;;;;..........;;~
2.5 -
E25
0-2
30
20
15
5
1
0.5
O0
0.5
1
1.5
2
2.5
3
35
4
4.5
Z (cm)
(a) Instantaneous ion temperature, parallel to the magnetic field lines.
2.5
tI"
1.5
0.5
'0
0.5
1
1.5
2
2.5
Z (cm)
3
3,
4
4.5
(b) Instantaneous ion temperature, perpendicular to the magnetic field lines.
Figure 3-11: Ion temperature with respect to magnetic field directionality.
~;;;;;;;;;~;
_;;;;;
~;;;;;;;;;~;;;;;;;;;;;~---------------~
3.5
1
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
Z (cm)
(a) Instantaneous electron number density.
ni Icm 3
6.5 E+13
2.7E+13
1.1 E413
4.7E+12
E
r
1.9 E+12
2.5
8.1 E+11
3.4 E+ 11
1.4E+1 1
2
5.8 E410
2.4E+10
1.0E4 10
1.5
1
0.5
0
0
0.5
2.5
z (cm)
(b) Instantaneous singly charged ion number density.
Figure 3-12: Electron and ion number densities.
4.5
;
............
3,5
;
;
;
;
~
D v)
449.9
405.0
360.0
315.0
270.01
225.0
2.5
E
180.0
1135.0
90.0
45.0
0.0
CC' 2
1.5
0,5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Z (cm)
th
(a) DCFT instantaneous potential distribution for the 350,000 iteration, with
plasma present.
449.9
3 -
2.5 MOM
E
2
1.5 -
405
360
315
270
225
180
135
45
0
0,
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Z (cm)
(b) Vacuum potential of DCFT. This potential distribution is present during the
neutrals only simulation referenced in Section 3.4.
Figure 3-13: The instantaneous potential distributions with and without plasma present in
the DCFT chamber.
g;~:
iiiiiiiii~;~;_;;::;;;;;;;;;;;~ :_~eiii_~_Li~::
~iiiiiii~i~iiiiiii~
-
2
7500
'3750
-150OOO
1,5
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
4
4.5
Z(cm)
(a) The axial component of the electric field.
3.5
2.5
0.5
0
0.5
1
1.5
2
2.5
3
3,5
Z(cm)
(b) The radial component of the electric field.
Figure 3-14: The gradient of the scalar potential from Figure 3-13(a), for the 350,000th
iteration.
44
Chapter 4
Conclusions and Future Work
Simulations of the performance and plasma dynamics of the DCFT were unable to be
completed for the case of full magnetic field strength (B). However, reducing B by a factor
of 5 allowed a convergent simulation to be run, which is an important intermediate step
toward simulating a real DCFT. Although the results presented here indicate that anode
specific impulse and thrust would not change by more than a few tens of a percent as a
result of the weakened magnetic field, the anode thrust efficiency would drop by nearly
50%. Weakening the magnetic field by a factor of 5 increases the mobility of electrons to
the point that the anode current could be expected to double. As simulated, the DCFT was
running in the high current mode, characterized by the presence of two distinct ionization
zones; one located between the anode and the upstream cusp, and the other located between
the two cusps[1]. Furthermore, insights into the basic operating principles are possible by
examining the trajectories of ions and electrons within the DCFT chamber.
4.1
DCFT Operating Principles
In order to understand the basic operating principles of the DCF thruster, it will be important to determine the roll that azimuthal currents play in determining the performance
of cusped-field thrusters. To reach this task, it will be beneficial to establish what kinds of
currents are present and what their causes are. The most obvious candidates are guiding
center drifts that arise when real or effective forces are applied to charged particles moving
through magnetic fields. Identifying candidate drift phenomena, as well as determining
where a species is magnetized, will allow insight into which drift velocities are the most
significant within a given region of the thruster.
4.1.1
Electron and Ion Drift Velocities
Hall effect thrusters take their name from the Hall current, an azimuthal electron current
that arises as magnetized electrons move under the combined influence axial electric and
radial magnetic fields. Their guiding center drift velocity is given by Equation 4.1, and
plotted for the DCFT simulated at 1/5 magnetic field strength in Figure 4-1(a).
The
direction of the drift does not depend on the charge of particle, and hence, if both species
are magnetized within a given region, no net current is established. Therefore, the lack of
magnetized ions is essential to the operation of Hall effect thruster.
ExB
=
B2
(4.1)
Although lacking prevalence in Hall thrusters, the drifts related to magnetic field topology can also give rise to currents within plasmas. Unlike the Hall current, these drifts,
referred to as gradient-curvature drifts, drive currents even when ions and electrons are
both magnetized[23].
However, the magnetization current that arises due to nonunifor-
mities in the magnetization of a plasma often have a canceling effect on current densities
arising from gradient-curvature drifts (Martinez-Sanchez, M., personal communication, August, 2009). A simple demonstration comes from drawing a current through a long straight
wire surrounded by a plasma. The currents that would occur based solely on gradientcurvature drifts are canceled out exactly by the magnetization current, given in Equation
4.3. Therefore, devices like the DCFT, having magnetic field strengths much larger in magnitude than Hall thrusters of similar size, could conceivably operate by some mechanism
other than the Hall effect. In fact, the regions within the thruster chamber where ions are
magnetized will determine where it is not possible for Hall currents to establish themselves.
Gc
(KE + -1
I 1 KE__
V
1
q
jM",s
Vx
B
2
,
(
xV(B 2)
(B
B4
B22
BKE
B
(4.2)
(4.3)
The azimuthal velocities for ions and electrons shown in Figures 3-8(c) and Figures 3-9(c)
show the direction changing before and after each magnetic cusp. For electrons, this change
is observed over the entire radial distance of the thruster near the upstream and downstream
cusps. The ions, however, exhibit the change in direction of the azimuthal velocity much
closer to the chamber walls, where the magnetic field is strongest. This suggests that the
ions are indeed magnetized within the DCFT.
Comparing the structure of the azimuthal velocities of ions and electrons with Figures
4-1(a) and 4-1(b), only the gradient-curvature drift velocity can account for the structure
and direction of ions and electrons in the cusps based the simulated data generated in this
study. However, upstream cusp nearest to the anode (z i 1 cm), the ExB drift provides a
more similar picture of the azimuthal electron current density distribution, shown in Figure
4-2. Although evidence suggests that ions are magnetized within some regions of the DCFT,
there contribution to the azimuthal current density is negligible compared to the electron
current in the same direction. Noticing that the ExB and gradient-curvature magnitudes
are both proportional to I/B, the magnitude of the azimuthal velocity of ions and electrons
will be lower in the actual DCFT as compared to the magnetic field simulated here.
4.1.2
Ion and Electron Magnetization
Figure 4-3 corroborates results presented in Figure 3-8(c) suggesting that ions are magnetized within the DCFT. In fact, this study has shown that, even with 1/5 the magnetic
field actually present in the design studied here, a significant portion of the DCFT ion
population is magnetized. The degree of magnetization in the full field DCFT should be
higher than the case simulated here. The values of the ion Hall parameter (Qi) presented in
Figure 4-3 were calculated by determining the ion cyclotron frequency based on the magnetic field data file used as input to this simulation, and by estimating the value of the ion
collisional frequency(ve,i). The magnetization profile visible in Figure 4-3 may explain the
hollow distribution of energetic ions within the plume, as measured previously[6].
The ion collision frequency can be estimated by determining the cross sections for various
types of collisions, and then calculating the contribution from each collision type individually. Following the method used by Szabo, the collisions included in this estimate of the
total ion momentum transfer frequency are ion-neutral charge exchange, ion-neutral elastic
scattering, and ion-electron coulomb interactions[14].
The cross section for each type of
-
'Vo
1000
2.5 -
E
iiii
-------------------------------
-
o
-100
-50000
,2
-00000
1
1.5 _
-500000
0.5
0.5
0
1
2
1.5
2.5
3
3.5
4
4.5
Z ([cm)
(a) Calculated ExB drift velocity magnitude. The electric and magnetic fields are
assumed to have no azimuthal components. The ExB drift speeds were determined
by calculating the cross product of the electric field shown in Figures 3-14(a) and
3-14(b) with the static magnetic field shown in Figure 3-3
3
15000
5000
12jBxv(B
0125o
2.5
MP%
E
B4
)
(T
)
Z (cm)
2.5
-2500
-5000
-15000
1.5
0.5
00
0.5
1.5
3.5
4.5
(b) The gradient-curvature drift is proportional to the quantity plotted here. The
complete expression is provided in Equation 4.2.
Figure 4-1: Calculated azimuthal velocities for ExB drifts, and the variation of the term
depending on magnetic topology in the gradient-curvature drift.
collision are given in Equations 4.4, 4.5 and 4.6.
2
Qcex = [-.88211n c,. + 15.1262]2 x 10 16 cm , c, expressed in m/s
(4.4)
.
..
.............
3.5 -
," \cn2
50
1.
-2.5 0.0.5
9,2
00
1
o.1
-O.5
-1
1
0.5
1.5
2
2.5
3
3.5
4
4.5
Z (cm)
shown previously in FigFigure 4-2: Azimuthal current density, calculated from quantities
ures 3-9(c) and 3-12(a).
Qes
8.28072 x 10- 10
8.28072 x 10-10
,
c expressed in cm/s
(4.5)
Cr
S=6.5
2
10-1lnA
E
m2, E expressed in eV
(4.6)
(Qes), and ion-electron
The ion-neutral charge exchange(Qcex), ion-neutral elastic scattering
4.7 to complete the calcucoulomb (Qc) cross sections were then combined with Equation
4.5, cr was assumed to be the ion
lation of the ion collision frequency. In Equations 4.4 and
as the average relative energy of
thermal speed, V kmie. In Equation 4.6, E is approximated
(kTe)
electrons with respect to ions, whose energy can be neglected:
vc,i = neQcfi,e + nn (Qcex + Qes) Vi,n
(4.7)
thermal speed for a Maxwellian
In Equation 4.7, Vi,e is assumed to be the mean electron
~
~
~
-;;;;
/
distribution
Vrme
~'~'U7'-
~
;~""
11;;
';;
... .....
;;....~
), and vi,n is assumed to be the ion sonic speed,
Te)
\im7r'U
3.5
2.5
E
1.5
0.5
0
0
1
1,5
2
2,5
3
3,b
4
4,5
Z(cm)
calculated as the
Figure 4-3: DCFT Ion Magnetization Assessment: Ion Hall Parameter,
cross sections used
ratio of ion cyclotron frequency to ion collision frequency. The collisional
Ion-Neutral
Exchange,
Charge
to calculate the ion collision frequency were: Ion-Neutral
collisions
Elastic Scattering, and Ion-Electron Coulomb. The effects of wall recombination
close to the wall.
were ignored. Including ion wall collisions would reduce the Hall parameter
reduced by
strength
field
magnetic
These results are from a simulation completed with the
a factor of 5.
species is presented
An alternative representation of the degree of magnetization for each
as magnetized when
in Figures 4-4(a) and 4-4(b). Heuristically, a particle may be thought of
or features containing
its Larmor radius is much less than the dimensions of the device
that the collisional
it. However, this picture of magnetization ignores the significant fact
of magnetization
frequency of a plasma must also be considered when determining the degree
of a species within a plasma. If a particle collides with other particles, or even boundaries,
able to execute the
at a frequency much larger than its own gyrofrequency, it will not be
parameter is of
drifts prescribed by Equations 4.1, 4.2 and 4.3. For this reason, the Hall
vital consideration when determining a species' degree of magnetization.
4.2
Future Work
The preliminary simulation results presented here provide a glimpse of the depth of simulated data that would become available for interpretation should a convergent simulation be
successfully executed for the full magnetic field strength of the DCFT. If further attempts
to run the full B case continue to fail, a new approach will be taken, such as making the
permittivity factor a function in the R-Z plane, or having it vary as a function of time.
Allowing -y to vary from iteration to iteration might overcome the challenges associated
with the distinct transient behavior of the DCFT model, during which the number of simulated particles increases dramatically before leveling off. Even without accomplishing the
simulation of a full field strength DCFT, insights into basic operating DCFT principles
demonstrate the potential benefit to be gained by further computational studies. Using
the data already generated in this model, a comparison between simulated ion flux to the
chamber walls with existing experimental data [11] could aid in interpreting those data and
offer some quantitative evidence concerning the effectiveness of DCFT erosion mitigation,
and help guide future design efforts to create a lower power, longer life alternative to Hall
thruster technology.
~i
i
i
i~
.
2.5 -
E
2
V) 1.nYz
0.002
0.0015
S2
0.001
M:
illll-
~
llII~i
:,iiiiiiiiii~i
~ ~;;;......................;;;;;;
0.0005
0.0001
:
1E-05
1.5
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
Z (cm)
(a) Spatial variation of average electron Larmor radius. In regions where
the electron density approaches zero (see Figure 3-12(a)), the Larmor
radius becomes a meaningless quantity. The perpendicular velocity was
taken as the bulk flow component orthogonal to the magnetic field vector.
3
r(m)
2.5
50
10o
L5
2
1.5
0.5
0.15
0.075
0.5
00
0.5
1
1.5
2
2.5
Z (cm)
3
3.5
4
4.5
(b) Spatial variation of average ion Larmor radius. The Larmor radius
is based on the component of the macroscopic ion velocity perpendicular
to the magnetic field line.
Figure 4-4: Average Larmor radii for electrons and ions, calculated based on macroscopic
flow velocities.
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